XV.] MOTION IN AN ORBIT OF ANY INCLINATION. 71 



Now 



^ ^ 



cos a) = cos 6 cos 0' 4- sin 6 sin 0' cos i = cos (0 6'} cos 2 - + cos (6 + 6'} sin 2 - , 



f 1 



sin cos 6' cos sin & cos i = sin (0 9'} cos 2 - + sin (6 + 0') sin 2 - . 



2i 



Hence we have the following expressions for the three forces : 

 u mV 



^ "*" /3 



- ? ~'J Fi 1 + cos 2(0- #')} cos j ^ + {cos 20 + cos 20'} 2 cos'^ sin 2 1 

 2 r 2 22 



+ {l + cos2(0+0')}sin 4 |l 



r = | -~ [ sin 2 (0 - 6>') cos j ^ + sin 20 . 2 cos 2 ^ sin 2 ^ 



Zi T 2* 2i 2i 



+ sin 2 (0 + 0') sin 4 ^1 

 *J 



Z = | ^ sin 1 1" - sin (0 - 20') cos 2 ~ + sin cos i + sin (0 + 20') sin 2 |l . 



Now we have seen 



di _ Zrcos0 dN _ Zrsm0 

 di ~ H ' ~dt ~~ ~ H sin i ' 



also, the rate of advance of the node along the orbit is 



Zr sin 

 H tan i ' 



Thus the equations of motion become 



Hd0 



r*~ dt H tan i 



d"r H* 

 together with - --= -P, 



dH 



